1
PHY 445/515
increased to the point where degeneracy between adjacent ∆MF transitions is broken, and use these results to determine the helicity of the circularly polarized light. The light is σ+ polarized. Similarly, using strong RF fields in the High B-field regime, we observe multi-photon transitions with ∆MF = 2.
Submitted: March 9th, 2015
Introduction:
Optical Pumping and Magnetic Resonance Michael S. Santana Partner: Andrew Kulesa Department of Physics, Stony Brook University
Abstract: Presented are the results of our experimentation of Optical Pumping and Magnetic Resonance, which was completed in the Advanced Laboratory at Stony Brook University’s Physics Department. A high frequency Rubidium lamp is used to optically pump electrons in 85Rb to the 52S1/2, F = 2, mf = +2 state and in 87Rb to the 52S1/2, F = 3, mf = +3 state by sending circularly polarized light, resonant at the D1 transition line (λ = 795 nm), through a sample cell of Rubidium atoms. Optical Pumping as well as stimulated emission to the subsequent lower mf levels are verified by observing the transmitted light read by a Silicon photodiode detector as a function of radio-frequency signal (RF). We show this transition holds for both isotopes. Using the absorption frequencies, we are able to note the cancellation of the ambient magnetic field with the Maxwell coils to measure the ambient magnetic field in the laboratory. Similarly, using the Optical Pumping/Magnetic Resonance mechanism, we measure the hyperfine splitting of Rubidium in the ambient field and in a range of applied magnetic fields (Low B-field/Zeeman regime) to determine the hyperfine Lande gfactors, gF, and compare them to quantum theory. For 87Rb we found gF = 0.487 ± 0.040 and Bearth = (44.295 ± 7.922) µT and for 85Rb we found gF = 0.324 ± 0.032 and Bearth = (42.698 ± 7.356) µT. We observe multiple transitions in a frequency sweep when the external magnetic field is
[Elementary Quantum Mechanics Theory] In elementary quantum mechanics, we solved the one electron case of the Schrodinger equation (Hydrogen atom) to obtain the wave functions (orbitals) and the corresponding energy levels: 𝜓𝑛𝑙𝑚 (𝑟, 𝜃, 𝜙) = 𝑅𝑛𝑙 (𝑟)𝑌𝑙𝑚 (𝜃, 𝜙) 𝐸𝑛 =
−𝑚𝑒 𝑐 2 𝛼 2 𝑍 2 2𝑛2
Where n, l, and m are integer quantum numbers (eigenvalues of operators that commute with the Hamiltonian), R(r) is the radial-wave function, and 𝑌𝑙𝑚 (𝜃, 𝜙) is the spherical harmonics. Multi-electron atoms’ frameworks can be constructed via the “hydrogenic” orbitals and energy levels given above, subject to the Pauli Exclusion Principle. The multi-electron atom is then arranged such that the lowest energy orbitals are filled first then the next higher level(s) is/are filled until all electrons have been allocated (aufbau principle). Although the ground state configuration of the Rubidium atom seems complicated (1s22s22p63s23p63d104s24p65s), since it only has one valence electron outside the innermost 36 electrons with angular momentum of net zero, the electronic structure is rather simple. Thus, when exciting and deexciting Rb’s electrons with external electromagnetic fields, we can consider their effects on only the one electron. However, we may not build a multi-electron wave function for Rb using the 1-electron configuration since
2 the orbitals and energies are not identical to those of the Hydrogen atoms, which is due to Coulomb interactions between the innermost electrons (for example, in Rb, the 5s and 5p levels would be degenerate in the Hydrogen atom, but are not degenerate in Rb). [Spin-Orbit Coupling] Quantum mechanics allows us to give an accurate description of matter to include splitting of the atomic energy levels of Rb at the nano-eV scale via perturbation theory. In the Coulomb-only description, the 52P1/2 and the 52P3/2 levels would be degenerate, although the two levels are actually split by approximately 0.03 eV by a perturbation called spin-orbit coupling. This perturbation serves as one component of the relativistic corrections to the Schodinger equation, which in addition to the other corrections are labeled “fine structure” [1]. Spin-orbit coupling is best understood as an effect due to the magnetic fields produced by orbiting particles that are charged. In the case of an electron’s rest frame, the nucleus orbits the electron producing a magnetic field. The electron’s magnetic moment energy in the magnetic field produced by the orbiting nucleus, B, is -μ⋅B. The spin orbit term is proportional to L⋅S (magnetic field is proportional to the orbital angular momentum, and the magnetic moment is proportional to its spin angular momentum). With spin-orbit coupling, since LZ no longer commutes with the Hamiltonian, ML is no longer a good quantum number. However, L⋅S = ½ (J2 –L2 – S2) since J = L + S. Therefore, J, L, and S, corresponding to their respective operators, are good quantum numbers since J2 (operator) commutes with the Hamiltonian.
[Hyperfine Splitting] The fine structure energy levels are further split by the hyperfine interaction, resultant from the spin of the nucleus. The nuclear spin angular momentum is coupled to the magnetic field produced by the orbiting electrons (similar to spin-orbit coupling). Since the magnetic moment of the nucleus, μI, is more than 1000 times smaller than that of the electron, the atomic energy levels are split by much smaller energy differentials. The hyperfine Hamiltonian is given by - μI⋅Be, and is proportional to I⋅J. The good quantum numbers are now I, J, F, and MF since the total angular momentum of the electron and the nucleus is F = J + I. The number of hyperfine levels for each state are calculated by the addition of angular momentum, such that for the ground state, 52S1/2, J = 1/2 and I = 3/2, the lower bound of total angular momentum is F = 3/2 – 1/2 = 1 and the upper bound is F = 3/2 + 1/2 = 2, with no integer steps in between. The same relation holds for 52P1/2, but for 52P3/2, J = 3/2 and I = 3/2 so the lower bound is F = 3/2 – 3/2 = 0 and the upper bound is F = 3/2 + 3/2 = 3 with two integer steps in between. Although we have refined Rubidium’s atomic structure down to approximately 10 nano-eV, this experiment calls for the measurement of even smaller splittings, so we must probe for additional effects on the atomic energy levels [1]. [Zeeman Effect] Although each hyperfine level is (2F+1)-fold degenerate (each hyperfine level contains 2F+1 magnetic sublevels, and the energy levels do not depend on the z-component of angular momentum since the atom is spherically symmetric), the situation varies when an external magnetic or electric field is applied. The Hamiltonian for the atoms interaction with the external magnetic field is given by μatom⋅Bext, where μatom is the total magnetic moment of the atom (sum of electron spin
3 angular momentum, electron orbital angular momentum, and nuclear spin angular momentum components), and is given by: 𝜇𝑎𝑡𝑜𝑚
−𝜇𝐵 = (𝑔𝑠 𝑺 + 𝑔𝐿 𝑳 + 𝑔𝐼 𝑰) ћ
If we consider that the external magnetic field is in the z-direction, the Hamiltonian becomes: 𝐻𝑍𝑒𝑒𝑚𝑎𝑛 =
𝜇𝐵 𝐵𝑒𝑥𝑡 (𝑔𝑠 𝑆𝑧 + 𝑔𝐿 𝐿𝑧 + 𝑔𝐼 𝐼𝑧 ) ћ
Although the Zeeman Hamiltonian indicates that we should use (MS, ML, and MI) as the quantum numbers to define angular momentum, and the Spin-Orbit Coupling and the Hyperfine Interaction indicate that we should use J, I, F, and MF as the good quantum numbers, it turns out that only the component of angular momentum that is parallel to the total angular momentum is of importance [1]. Thus, one can think of the components of J, S, and I precessing about the vector of total angular momentum, F (vector model), and averaged over time, the components along the vector F remain. This simplification allows us to write the Zeeman Hamiltonian in terms of the zcomponent of total angular momentum: 𝜇𝐵 𝐵𝑒𝑥𝑡 𝐻𝑍𝑒𝑒𝑚𝑎𝑛 = 𝑔𝐹 𝐹𝑧 (𝑊𝑒𝑎𝑘 𝐵 − 𝑓𝑖𝑒𝑙𝑑) ћ With the pre-factor being the hyperfine Lande g-factor, which is given by: 𝑔𝐹 𝐹(𝐹 + 1) − 𝐼(𝐼 + 1) + 𝐽(𝐽 + 1) = 𝑔𝐽 2𝐹(𝐹 + 1) 𝐹(𝐹 + 1) + 𝐼(𝐼 + 1) − 𝐽(𝐽 + 1) + 𝑔𝐼 2𝐹(𝐹 + 1)
And the fine structure Lande g-factor, 𝑔𝐽 is given by: 𝑔𝐽 𝐽(𝐽 + 1) − 𝑆(𝑆 + 1) + 𝐿(𝐿 + 1) 2𝐽(𝐽 + 1) 𝐽(𝐽 + 1) + 𝑆(𝑆 + 1) − 𝐿(𝐿 + 1) + 𝑔𝑆 2𝐽(𝐽 + 1) = 𝑔𝐿
We are to compare our experimental values of the g-factors to the accepted values found in the references by Steck [2, 3]. Evaluating this with the first order perturbation theory, we are yielded (Weak B-field): Δ𝐸 = ⟨𝐹, 𝑀𝐹 |𝐻𝑍𝑒𝑒𝑚𝑎𝑛 |𝐹, 𝑀𝐹 ⟩ Which grants us the two frequencies at which we expect to observe magnetic resonance in the earth’s magnetic field (~328 kHz and ~218 kHz) for both isotopes, respectively. If the interaction of the external magnetic field is greater than the hyperfine coupling, perturbation theory breaks down since the perturbation is no longer small compared to the Hamiltonian. Thus, it is possible that the external magnetic field applied is greater than the internal magnetic field that is responsible for hyperfine splitting [1]. If this is the case, |𝐹, 𝑀𝐹 ⟩ is no longer a good basis and I and J decouple. The good basis then becomes|𝐽𝑀𝐽 𝐼𝑀𝐼 ⟩ (Strong B-field): 𝐻𝑍𝑒𝑒𝑚𝑎𝑛 =
𝜇𝐵 𝐵𝑒𝑥𝑡 (𝑔𝐽 𝐽𝑧 + 𝑔𝐼 𝐽𝑧 ) ћ
Δ𝐸 = ⟨𝐽, 𝑀𝐽 , 𝐼, 𝑀𝐼 |𝐻𝑍𝑒𝑒𝑚𝑎𝑛 |𝐽, 𝑀𝐽 , 𝐼, 𝑀𝐼 ⟩ Figure 1 shows splitting of the ground state of 87 Rb in both the weak field regime (Zeeman regime) and the strong field regime (PaschenBack regime), where the dimensionless parameter helps us to define what is meant by a “strong” versus a “weak” field, and is given by: 𝑥≡
(𝑔𝐽 − 𝑔𝐼 )𝜇𝐵 𝐵𝑒𝑥𝑡 Δ𝐸𝐻𝐹
4 Where x > 1 represents the Paschen-Back regime. In the ground state of Rb (J = 1/2), the intermediate region can be determined with the solution of the Breidt-Rabi equation.
Fig. 1. 87Rb 52S1/2 (ground) level hyperfine structure in an external magnetic field. The levels are grouped according to the value of F in the low-field (Zeeman) regime and mJ in the strong-field (hyperfine Paschen-Back) regime. Taken from [2]. Experimental Details and/or Methods:
Fig. 2. The apparatus for the Optical Pumping and Magnetic Resonance experiment. Taken from [1]. Radiation from the Rb lamp (D1 and D2 lines) is culminated in a lens. The filter selectively chooses the D1 light, which is sent through a polarizer and quarter wave-plate making the light circularly polarized. The helicity of the light is determined in the high B-field regime (σ+).
The Rubidium cell contains both isotopes of Rubidium (87Rb and 85Rb) as well as 20 Torr of Argon, which acting as a buffer gas, prevents Rb atoms from hitting the walls of the cell too frequently, which would depolarize them. Light that makes it through the Rubidium cell is then refocused by another lens onto a Silicon photodiode, which is reversed biased with 9V battery, and is in series with a 300 kΩ resistor. The resistor changes photocurrent from the photodiode to voltage, which is read out by a HP 34401A multimeter and a computer. The mechanism is tilted to be approximately parallel to the earth’s magnetic field. The cell is encompassed by a set of Maxwell coils, which when a current is applied to them, a uniform magnetic field through the cell is produced. Helmholtz coils are similarly encompassing the cell, but are orthogonal to the Maxwell coils. The Helmholtz coils provide resonant radio frequency fields which act to stimulate emission of the previously spontaneous (slow) relaxation rate of excited magnetic sublevels (sweeping the synthesizer frequency across resonance). Although Rb is shown to be a solid on the periodic table, it has a relatively high vapor pressure, which is increased substantially with an increase in temperature. Using the heating gun, which is controlled by an AC transformer, more Rb is allocated to the gas phase. The formula for pressure as a function of temperature is in the Steck article(s) [2, 3]. The temperature is read out by the thermometer sticking out the backside of the Rb cell. [Methods] One of the necessary conditions for an atom to make a transition (transitions caused by incident electromagnetic radiation) is that the incident light frequency must be resonant with the atomic transition. Other necessary conditions that must be satisfied are the “selection rules,” which are responsible for optical pumping and allow us to measure
5 Zeeman splitting. For example, assuming the helicity of the circularly polarized light is σ+ polarized, ΔMF = +1. An atom in the 52S1/2F = 1, MF = 0 state of 87Rb can only make a transition to the 52P1/2 F = 2, MF = 1 state. The excited P states undergo spontaneously emission in a matter of nanoseconds to the 52S1/2 state and are emitted as unpolarized, causing transitions with ΔMF = 0, 1, or -1. In other words, the firstorder transitions between adjacent Zeeman levels are magnetic dipole transitions, for which the selection rules are ∆l = ±1, ∆j = ±1, and ∆m = −1, 0, 1. In particular, right-handed circularly polarized light will induce ∆m = +1 transitions and left-handed circularly polarized light will induce ∆m = -1 transitions [5]. Thus, when we excite an atom with D1 light, we take one step up the MF ladder, but have a random process upon emission. Atoms that end up in the 52S1/2 F = 2, MF = 2 level remain optically pumped when excited since there is no 52P1/2 F = 2, MF = 3 level for them to be excited to [1]. This phenomena is illustrated in Figure 3 below:
excitation with the pumping light is not allowed, unless the atoms are “dumped” down to lower MF levels through stimulated emission with an RF field. Taken from [1]. Since these optically pumped atoms cannot absorb the D1σ+ light, they remain pumped up at this state (spontaneous emission is very slow in the absence of collisions), and are considered to be in a metastable state. Moreover, since all the atoms are optically pumped up and cannot absorb the incident D1σ+, the medium becomes transparent and the light passes through them. Although the spontaneously relaxation rate is relatively slow, it can be made faster (stimulated emission) by applying a resonant radio-frequency field. This process is called magnetic resonance, or double resonance spectroscopy, which is the primary measurement in this experiment: measuring transmission of D1σ+ light versus the resonant radio-frequency field. If the (RF) is resonant, a dip is observed in the light transmitted by the medium signifying a medium transition from transparent to opaque. By calculating the center of the dip or peak (fitting methods), the Zeeman sub-level splitting can be determined. Results: In the ambient magnetic field in the laboratory, we observed magnetic resonance approximately near the frequencies predicted when evaluating the first order perturbation theory for the two isotopes (Weak B-field). Below I have fit 87Rb to both a Gaussian and Lorentzian profile to see which fit is more appropriate.
Fig. 3. Illustration of optical pumping in 87Rb with σ+ light. The upward solid arrows correspond to transitions induced by the circularly polarized pumping light from the lamp. The downward dashed lines correspond to spontaneous emission. When atoms reach the MF sublevel of the ground state, further
Photodiode Signal (V)
6
Magnetic Resonance Peak of 87Rb in Ambient Field 0.297 0.2965 0.296 0.2955 Data 0.295 0.2945 Gaussian 0.294 Fit 0.2935 270 290 310 Resonance Frequency (kHz)
Photodiode Signal (V)
Fig. 4. The Gaussian fit yields an X2/DOF of 0.017744, which proves to provide a worse description of the data than the Lorentzian fit.
0.297 0.2965 0.296 0.2955 0.295 0.2945 0.294 0.2935
Magnetic Resonance Peak of 87Rb in Ambient Field
provided by the Maxwell coils. The Maxwell coils are arranged in a configuration such that it consists of three coaxial coils: the largest, central coil A as well as the two smaller coils B and C adjacent to A, with a distance between A and B and A and C along the main axis equal to z. Assuming the coils are infinitely thin, the magnetic field due to the Maxwell coils on its main axis is described by the Biot-Savart Law [5]: 𝐵𝑐𝑜𝑖𝑙 = 𝜇0 𝐼 (
𝑁𝐵,𝐶 𝑟𝐵,𝐶 2 (𝑟𝐵,𝐶 2 + 𝑧 2 )2
+
𝑁𝐴 𝑟𝐴 2 3
)
2𝑟𝐴 2
𝐵𝑐𝑜𝑖𝑙 = 0.000421 [𝑇/𝐴] 𝐼 With an error of: 2
Data Lorenzian Fit 270
290
310
Resonance Frequency (kHz)
Fig. 5. The Lorentzian fit yields an X2/DOF of 0.004384, which proves to provide a better description of the data than the Gaussian fit. When fitting other dips for magnetic resonance, we will employ the same fitting procedure of fitting to Lorentzian profiles to determine the location parameter. Microsoft Excel’s Solver Package was used to optimize the fits, setting the objective function to minimize X2, the parameters to be varied, and the appropriate constraints similarly. [Calculation of Bearth and gF] Using the Optical Pumping/Magnetic Resonance mechanism, we measure the Zeeman splitting of 87Rb and 85Rb in both the laboratory’s ambient magnetic field as well as in a wide range of applied external magnetic fields
Δ𝐵𝑐𝑜𝑖𝑙
𝜕𝐵 𝜕𝐵 2 = √( ) 𝜎𝑟𝐵,𝐶 2 + ( ) 𝜎𝑟𝐴 2 𝜕𝑟𝐵,𝐶 𝜕𝑟𝐴 𝜕𝐵 2 2 𝜕𝐵 2 2 + ( ) 𝜎𝑧 + ( ) 𝜎𝐼 𝜕𝑧 𝜕𝐼
The above relation for Bcoil/I allows for the calculation of magnetic field for a given current. A more precise measurement for Bcoil would take into account the thickness of the coils, and would utilize a script to sum the individual contribution that each layer in each coil have to the magnetic field at the Rb cell, which is at the center of the coil “A,” as a function of applied current [6]. We did not code such a script in the interest of time, and accepted our value for Bcoil/I to be appropriately accurate. Using the weak field equation, we relate a range of magnetic fields applied by the Maxwell coils to the Zeeman splitting resonance frequencies for both isotopes of Rb. Similarly, we change the direction of the current (changing the direction of the magnetic field), by swapping the alligator clips on the back of the power supply and relate this range of fields
7 applied by the coils to the Zeeman splitting resonance frequencies.
Resonance Frequency (Hz)
Resonance Frequency vs. Applied Magnetic Field for 87Rb and 85Rb 600000 500000 400000 300000
Linear (87Rb Positive) Linear (85Rb Positive)
200000 100000 0 0.000030.000050.000070.000090.000110.00013
Applied Magnetic Field (T)
Fig. 6. Displayed above are the plots of resonance frequency vs. applied magnetic field for both isotopes of Rubidium. The negative equivalents are omitted from the plot, but were observed and used to verify the results of the positive field regime. The expressions for 85Rb and 87Rb (positive and negative) were fit to a linear relation, and their respective slopes were adjusted to yield the hyperfine Lande g-factors, gF. Similarly, we solved for the x-intercept of each fit, which by cancellation of the ambient field with our Maxwell coils yielded the ambient magnetic field in the advanced laboratory. For 87Rb we found gF = 0.487 ± 0.040 and Bearth = (44.295 ± 7.922) µT, and for 85Rb we found gF = 0.324 ± 0.032 and Bearth = (42.698 ± 7.356) µT. The accepted values for the hyperfine Lande gfactors are gF = 1/2 and gF = 1/3 for 87Rb and 85 Rb respectively, for which both of our experimental g-factors were in agreement with theory within their error(s) [2, 3]. The accepted value of the earth’s magnetic field at our precise latitude and longitude was given to be 51.8944 µT, for which 87Rb was in agreement within its respective error though 85Rb was not [4]. Variation in the accepted value for earth’s magnetic field as compared to the ambient field
measured in the advanced laboratory are likely caused by the surrounding building walls and/or magnetic fields produced by current(s) within electronics. The values for the x-axis (B-field) were determined from the scale factor previously determined. Error was assigned to each data point by means of the relation derived above for ΔBcoil. Resonance frequencies were obtained at a range of applied magnetic fields for both isotopes by fitting each magnetic resonance dip with a Lorentzian profile, and choosing the optimized location parameter (center frequency). Error was calculated for each resonance frequency by varying the location parameter until it deteriorated the reduced chisquared such that X2 -> X02+1, where X02 is the reduced chi-squared (X2/DOF). Although it may have been more precise to re-fit after each change to the location parameter, it was not necessary. [Calculation of Absorption Cross Section] We recorded the lamp cell and the Rb cell warmup curves (Voltage vs. Temperature) in an effort to determine the absorption cross section of our optically pumped Rb atoms. We used the Beer-Lambert Law, which relates the attenuation of light to the properties of the material through which the light is traveling, which is given by: 𝐼 = 𝐼0 𝑒 −𝑛𝜎𝑙 Where I is the radiant flux transmitted by the material, I0 is the radiant flux received by the material, n is the number density of the attenuating species in the material, σ is the attenuation cross section, and l is the path length [7]. Dividing I by I0, and then taking the natural log of both sides, we obtain an expression given by: ln(𝐼) = ln(𝐼0 ) − 𝜎(𝑛𝑙)
8 If we plot this relationship with ln(I) on the yaxis vs. nl on the x-axis, the slope is equal to the absorption cross section, σ, in units of area. In order to obtain n, the number density of the attenuating species in the material, we must use the Ideal Gas Law, which is given by: 𝑃𝑉 = 𝑁𝑘𝐵 𝑇 𝑃 = 𝑛𝑘𝐵 𝑇 𝑛=
𝑃 𝑘𝐵 𝑇
To calculate n, we must know P, although currently we only know T. Therefore, we use the vapor-pressure model from Steck references [2, 3]: 𝑙𝑜𝑔10 𝑃𝑉 = 2.881 + 4.857 −
4215 (𝑠𝑜𝑙𝑖𝑑) 𝑇
Where pressure, P, is in torr and temperature, T, is in Kelvin. We are at last able to solve for n, and thus the absorption cross section, σ, as well as I0 if desired (the radiant flux received). ln(I) vs. nl (Absorption Cross Section) -1.12
ln(I) (ln(Volts)
-1.14 0
5E+15
-1.16 -1.18 -1.2 -1.22 -1.24 -1.26 -1.28
nl (# of molecules/m2)
Fig. 7. After fitting the data to a line of the form y=mx+b, we used the solver to minimally optimize the chi-squared, and the absorption cross section was found to be equal to σ = 6E17m2.
[High B-field Regime and Multi-Photon Transitions] When the magnetic field becomes strong enough, as does RF power, degeneracy splits, making it possible to resolve the transitions for each ΔmI state given an mJ value. In the case of 87 Rb, we hope to show two photon transitions with ΔmI = -2. In the strong field, with high RF power, mI splittings are large. Electrons can deexcite to a lower mI level instead of being optically pumped to the 5p2P1/2, and then can again deexcite to the next mI level (undergoing another magnetic resonance transition) such that mI(final) = mI(initial) – 2 (equivalent 3 level system). For 85Rb I = 5/2, mJ = +1/2, -1/2 and mI = -5/2, -3/2, -1/2, +1/2, +3/2, +5/2, totaling 5 possible transitions. For mJ = +1/2, states with mI > 0 have higher energies than states with mI < 0. The opposite is true for mJ = -1/2. For 87Rb I = 3/2, mJ = +1/2, -1/2 and mI = -3/2, -1/2, 1/2, 3/2, totaling 3 possible transitions. For mJ = +1/2, states with mI > 0 have higher energies than states with mI < 0. The opposite is true for mJ = -1/2. For 85Rb I = 5/2, we expect the +5/2 -> +3/2 transition to have the strongest signal at the lowest energy, with each lower transition decreasing in signal strength but increasing in energy separation. For 87Rb I = 3/2, we expect the +3/2 -> +1/2 to have the strongest signal at the lowest energy, with each lower transition decreasing in signal strength but increasing in energy separation [6]. In the below plots, we exclusively include the high B-field/high RF power regime for 85Rb, although we did observe the presence of transitions between the mI states and the correct number of transitions (excluding the unexpected dip), as well as multiphoton transitions.
9
85Rb,
85Rb,
I=-0.8489A Photodiode Signal (V)
Photodiode Signal (V)
0.299 0.2988 0.2986 0.2984 0.2982 0.298 0.2978 1750
1800
1850
1900
Frequency (kHz)
I=-1.0997A
Photodiode Signal (V)
0.2991 0.299 0.2989 0.2988 0.2987 0.2986 0.2985 0.2984 2250
2300
0.2989 0.2988 0.2987 0.2986 0.2985 0.2984 0.2983 2450
2500
2550
2600
Frequency (kHz)
Fig. 8. The main dip is around 1820 kHz. The splittings between the lower mI states are shown to be approximately 1827, 1834, and 1841 kHz respectively. The weaker dips occurring at higher energies indicate electrons going from one mI state to the one below. 85Rb,
I=-1.1997A
2350
2400
Frequency (kHz)
Fig. 9. The main dip is around 2280 kHz. The presence of transitions between the mI states is now definitely observable, with splittings between lower mI states to be approximately 2290, 2301, 2312, and 2326 kHz respectively. Similarly, multi-photon emission is noted at 2285 and 2296 kHz respectively. There is an unexpected dip at approximately 2341 kHz.
Fig. 8. Displayed is a frequency sweep for 85Rb at a High RF field in the High B-field regime. We continue to observe both multiple transitions between mI states as well as multiphoton transitions. The energy spacing increases between the transitions and the number of dips is consistent with theoretical predictions, which is reassuring (excluding the unexpected dip at 2537 kHz). Due to the orientation of the transition with the strongest signal at the lowest energy, with respect to each lower transition, decreasing in signal strength and increasing in energy separation (to the right of the strongest signal with the lowest energy), we conclude that the helicity of our D1 pumping light is σ+ circularly polarized. [Lineshape Broadening Considerations] Although we did not make any direct calculations involving lineshape broadening, it is important to note the significance of its effects. The following broadening effects spurred my interest, and would make for interesting exercises; Natural Broadening: The broadening of a spectral line due to the finite lifetime of an excited state of an atom, which is in accordance with the uncertainty principle such that: ∆𝐸∆𝑡 ≳ ℎ
10 Where ∆t = τ, and broadening is equal to: ∆𝐸 ∆𝑣 = ≳ 𝜏 −1 ℎ
∆𝑣𝐹𝑊𝐻𝑀 ≅ 𝜏
−1
𝑃 1/2 (1 + ) 𝑃𝑠𝑎𝑡
Conclusion:
This type of broadening is intrinsic to the transition, and the spectral line is best described by a Lorentzian profile, where the resonant frequency (center of the dip) is v0 with a FWHM of τ-1 [8]. Collision Broadening: In the lamp, Rubidium atoms often collide with the walls of the container and with each other respectively. These collisions can shorten the lifetime of the excited state provided that (τcollision < τ), which results in the Kinetic Theory relation: 1
1 𝜋𝑚𝑘𝑇 2 𝜏𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛 ~ ( ) 𝜎𝑆 𝑃 8 This means that broadening is proportional to pressure. Doppler Broadening: Random thermal variations and movements of atoms cause Doppler shifts in observed frequencies, which serves as an additional source of line broadening. Included in a dip are the atoms that are moving at right angles with respect to the observers, atoms that are moving away from the observer, and atoms that are moving towards the observer. The Doppler width, ∆vD is given by: 2𝑢 ∆𝑣𝐷 ≅ 𝜆 Where u is the average atomic velocity, and via Kinetic Theory, u = (2kT/MRB) ½ [8]. Power Broadening: If a system is saturated (two atomic levels have equivalent populations), then the power of the RF field at which this happens is denoted by Psat. The FWHM of the Lorentzian is given by a function dependent on power, P:
In this experiment we successfully demonstrated applications of optical pumping, and its role in the preparation of states for magnetic resonance. The linear relationship of resonance frequency vs. applied magnetic field for both 85Rb and 87Rb yielded accurate hyperfine Lande g-factors, and allowed for the calculation of the ambient magnetic field in the advanced laboratory. In the high B-field/high RF power regime, the presence of transitions between the mI states is observable, as is multiphoton transitions in between states. Lastly, we conclude that the helicity of our D1 pumping light is σ+ circularly polarized. Acknowledgments: We would like to thank Professor Allison for his willingness to thoroughly answer any and all questions we had in laboratory. References: [1] Professor Allison, Optical Pumping and Magnetic Resonance (2014). [2] D. A. Steck, “Rubidium 85 d line data, available online at http://steck.us/alkalidata (revision 2.1.4, 23 December 2010).” [3] D. A. Steck, “Rubidium 87 d line data, available online at http://steck.us/alkalidata (revisio”n 2.1.4, 23 December 2010).” [4]“http://www.ngdc.noaa.gov/geomag/wmm/ dodwmm.shtml”. [5] D.C. Elton and J. Chia-Yi, “Optical Pumping of Rubidium Vapor, available online at http://mysbfiles.stonybrook.edu/~delton/Articl es/LATEX/Lab1_Optical_Pumping_Dan_Elton.pd f (27 April 2012).”
11 [6] James Dragan and Stefan Evans, Optical Pumping and Magnetic Resonance (4 October 2013). [7]”http://en.wikipedia.org/wiki/Beer%E2%80% 93Lambert_law”. [8] Adam Esmail, Optical Pumping in Rubidium (24 January 2011).